Grothendieck universe

From Exampleproblems

In mathematics, a Grothendieck universe is a set with the following properties:

1. If x ∈ U and if y ∈ x, then y ∈ U.
2. If x,y ∈ U, then {x,y} ∈ U.
3. If x ∈ U, then P(x) ∈ U. (P(x) is the power set of x.)
4. If is a family of elements of U, and if I ∈U, then the union is an element of U.

A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, it provides a model for set theory.) As an example, we will prove an easy proposition.

Proposition 1.

If x ∈ U and y ⊆ x, then y ∈ U.

Proof.

y ∈ P(x) because y ⊆ x. P(x) ∈ U because x ∈ U, so y ∈ U.

It is similarly easy to prove that any Grothendieck universe U contains:

• All singletons of each of its elements,
• All products of all families of elements of U indexed by an element of U,
• All disjoint unions of all families of elements of U indexed by an element of U,
• All intersections of all families of elements of U indexed by an element of U,
• All functions between any two elements of U, and
• All subsets of U whose cardinal is an element of U.

In particular, it follows from the last axiom that if U is non-empty, it must contain all of its finite subsets and a subset of each finite cardinality. One can also prove immediately from the definitions that the intersection of any class of universes is a universe.

Grothendieck universes are equivalent to strongly inaccessible cardinals. More formally, the following two axioms are equivalent:

(U) For all sets x, there exists a Grothendieck universe U such that x ∈ U.

(C) For all cardinals κ, there is a strongly inaccessible cardinal λ which is strictly larger than κ.

To prove this fact, we give explicit constructions. Let κ be a strongly inaccessible cardinal. Say that a set S is strictly of type κ if for any sequence s_n ∈ ... ∈ s₀ ∈ S, |s_n| < κ. (S itself corresponds to the empty sequence.) Then the set u(κ) of all sets strictly of type κ is a Grothendieck universe of cardinality κ. The proof of this fact is long, so for details, we refer to Bourbaki's article, listed in the references.

To show that the large cardinal axiom (C) implies the universe axiom (U), choose a set x. Let x₀ = x, and for all n, let x_n = ∪x be the union of the elements of x. Let y = ∪_nx_n. By (C), there is a strongly inaccessible cardinal κ such that |y| < κ. Let u(κ) be the universe of the previous paragraph. x is strictly of type κ, so x ∈ u(κ). To show that the universe axiom (U) implies the large cardinal axiom (C), choose a strongly inaccessible cardinal κ. κ is the cardinality of the Grothendieck universe u(κ). By (U), there is a Grothendieck universe V such that U ∈ V. Then κ < 2^κ ≤ |V|.

In fact, any Grothendieck universe is of the form u(κ) for some κ. This gives another form of the equivalence between Grothendieck universes and strongly inaccessible cardinals:

For any Grothendieck universe U, |U| is a strongly inaccessible cardinal, and for any strongly inaccessible cardinal κ, there is a Grothendieck universe u(κ). Furthermore, u(|U|)=U, and |u(κ)|=κ.

Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of Zermelo-Fraenkel set theory, the existence of universes cannot be proved from Zermelo-Fraenkel set theory either.

The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.

References

Bourbaki, N., Univers, appendix to Exposé I of Artin, M., Grothendieck, A., Verdier, J. L., eds., Théorie des Topos et Cohomologie Étale des Schémas (SGA 4), second edition, Springer-Verlag, Heidelberg, 1972.